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Finite, Spurious Infinite, True
Infinite

M. A.
Marchetti

A concise exposition of the development of the
true infinite is found in Hegel's Encyclopedia Logic (EL92-95).
It may be much easier to follow than the one given in the Science of
Logic. The following paragraphs are from the Gerates, et al translation of that
book, along with some parts of the "Additions" where I felt they were
useful. At the end I give my interpretation of the development.

EL§92
The being that is kept firmly distinct from the determinacy, being-in-itself,
would be only the empty abstraction of being. In being-there (Dasein) the
determinacy is one with being and is at the same time posited as negation; this
determinacy is limit, restriction. Thus, otherness is not something-indifferent
outside it, but its own moment. In virtue of its quality, something is first
finite and secondly alterable, so that the finitude and alterability belong to
its being.

Addition. In being-there
negation is still immediately identical with being, and this negation is what
we call "limit". Something only is what it is within its limit and by
virtue of its limit. We cannot regard limit, therefore, as merely external to
being-there; on the contrary, limit totally permeates everything that is there.
The interpretation of limit as a merely external determination of being-there
is based on a confusion of quantitative with qualitative limit. Here we are
dealing first with qualitative limit. When we are considering a piece of land
three acres in area, for example, that is its quantitative limit. But, in
addition, this piece of land is also a meadow and not a wood or a pond, and
this is its qualitative limit.

EL§93
Something becomes an other, but the other is itself a something, so it likewise
becomes an other, and so on ad infinitum.

EL§94This
infinity is spurious or negative infinity, since it is nothing but the
negation of the finite, but the finite arises again in the same way, so that it
is no more sublated than not. In other words, this infinity expresses only the
requirement that the finite ought to be sublated. This progress ad infinitum
does not go beyond the expression of the contradiction, which the finite
contains, [i.e.,] that it is just as much something as its other, and [this
progress] is the perpetual continuation of the alternation between these
determinations, each bringing in the other one.

Addition. If we let
something and other, the moments of being-there, fall asunder, the result is
that something becomes an other, and this other is itself a something, which,
as such, then alters itself in the same way, and so on without end.

EL§95
What is indeed given is that something becomes another, and the other becomes
another quite generally. In its relationship to an other, something is already
an other itself vis-a-vis the latter; and therefore, since what it passes into
is entirely the same as what passes into it—neither having any further
determination than this identical one of being an other—in its passing into
another, something only comes together with itself; and this relation to itself
in the passing and in the other is genuine Infinity. Or, if we look at it
negatively: what is changed is the other, it becomes the other of the other. In
this way being is reestablished, but as negation of the negation. It is now
being-for-itself.

The following is my
interpretation of the development, finalized with some additional comments
on the Concept.

We start with something, i.e. a finite thing - a flower, or any
finite thing or individual existence, or just the finite in general. The finite
has a being, but it also has a determinateness or quality.

The being of something is distinct from its determinateness. In
other words, the being of a flower and its scent are two distinct things. The
scent is a quality or determinateness of the being of the flower. The flower
and its scent are not identical because the flower is much more than just its
scent.

If we call the being of the flower its being-in-itself (or
being-by-itself), we indicate that being which is distinct and apart from its
determinate mode. In other words, being, before any determinations are made of
it, is pure being without any determinations, or simple undetermined being -
being in general. We will refer to this undetermined being as being-in-itself.

When a determination is made of this being-in-itself, such as the
scent of the flower - this quality or scent is something that belongs to the
flower - it resides in the being-in-itself of the flower. As being-in-itself,
the qualities that it possesses are not yet revealed or manifested - they are
implicit. But when being-in-itself is determined then its implicit qualities
are made explicit. After a determination of being-in-itself is made it then
becomes what is called determinate being.

So there is a difference between being-in-itself and determinate
being. Being-in-itself is indeterminate or abstract, while determinate being is
being with a qualitative determination. For example, a tract of land is
something indeterminate in that we know nothing about its particular qualities.
A tract of land may be a pond, a meadow, a forest, etc. The tract of land is
thus like being-in-itself until a qualitative determination is made and then we
know what type of land it is, which would be its determinate being.

In determinate being, the explicit determinateness is one with the
implicit being-in-itself. In other words, the tract of land is implicitly a
pond, let us say, before we explicitly determine it is a pond. It is not that
the thing-in-itself, may be anything different from what we determine it to be.
You might note here that this is Hegel's direct refutation of Kant, who claimed
we can't know anything about the thing-in-itself except its appearances. Hegel
claims that the appearances are only what the thing-in-itself or
being-in-itself is implicitly, which are made explicit when they are
determined. Kant artificially separates appearances from the in-itself or
implicit, while Hegel comprehends the unity in them.

The determinateness of a thing, however, is also different from
the thing-in-itself. The scent of a flower is NOT the flower. A tract of land
as a pond is different from tracts of land in general that are not ponds. When
the determinateness is considered in its difference from that being-in-itself
it is determining, i.e. when it is considered as determinate being as opposed
to being-in-itself, then it is considered a negation. Thus determination has
two sides: it is identical with what being-in-itself is implicitly, and at the
same time it is different from being-in-itself since it is just one
determinateness of that being.

Considered as an existent negation, determinateness is called
Limit. By calling a tract of land a meadow, it is limited to that qualitative
determinateness - it is not a pond or forest, etc. In this case, limit is not a
quantitative determination of size, etc., but a qualitative one, since we are
using determinateness as the quality of a thing.

In this way, the limit is something that is essential to that
which it limits- it is one with what it limits or determines. Therefore,
determinateness is both one with and different from its being-in-itself.
Likewise limit is one with what it is limiting and as its negation is different
from it. (EL92)

In general, something is what it is by its quality, and at the
same time something is different from its quality. Something as finite has its
limit, and this limit is the negation of the something or something else. So
finitude and variability both pertain to its being. The finite is alterable
within its own nature as both being and its negation of itself. It is therefore
both something and something else.

The negation or other of something is also something. That other
something also has a limit, or its negation, i.e. another something. In this
way the finite forms a progression of somethings, ad infinitum(EL93).
This is the bad or spurious infinite.

The finite is inherently contradictory. It is something and something
else. This establishes a progressive alteration showing that the finite as the
negation of itself is the infinite, but it is only the spurious infinite -
something that ought to be the infinite, but never reaches it because it always
retains its contradiction, the finite (EL94).

But the other of something is itself something of which its other
is the first something. So both are other to each other. In this otherness they
are identical. But that which passes into itself, which has itself outside
itself, is just the true infinite (EL95).

The finite has a limit, beyond which it does not find itself. But
the infinite has its limit inside itself, i.e. beyond its limit it finds only
itself. The infinite is thus the cancellation or negation of its own limit.
Since the limit is also a negation, the infinite is in this way the negation of
the negation.

We started with something (individuality). It is finite and has
some determinate quality. Its determinateness is distinguished from its
being-in-itself. This is the distinction between its particularity
(determinateness) and universality. This shows itself to be the relation
between something and something else, which leads to the spurious infinite. But
determinate being is not only something different from being-in-itself since it
is implicitly contained within being-in-itself. This identity within difference
between the two establishes the infinity of the unity of the universal and
particular.

This infinity is a result arrived at by the process of negation of
the negation - it does not belong to the immediacy of things. The determinate
is the negation of being-in-itself, or the particular is the negation of the
universal. The finite is what is negatively related to the infinite. In each
case the finite, particular or determinate is IN the infinite, universal or
implicit being. At the same time they are Other than the infinite, universal or
being-in-itself. This establishes their relationship as identical and
different.

The part of their identity is due to the negation of the Otherness
(which is itself, negation), i.e. the negation of the negation. This negation
of otherness, which results in the identification of the Other with itself, is
called Being-for-self. In being-for-self the otherness or determinateness is
acknowledged as a negation but it is negated as otherness or as independent of
the being that is determined. It is this negation of the independence of the
being of the Other that is referred to as being-for-self. This arises directly
out of the relation of the infinite as the negation of the negation. Because it
involves negation, it does not produce an abstract identity of one thing and
another - the resultant unity or truth must be understood as the result plus
the process for arriving at it. The original and final stages of the
development are different, even though there is a circular return involved in
being-for-self. The negation of the negation is a return to the original being,
but it includes its absolute negativity, without which it is mere abstract
being without negativity.

The concrete universal is that which contains its particularity
within itself, where the particulars are also different from it. These
particulars, it should be noted, are just universals when they are considered
independent of the universal. They are moments of universality when considered
in their unity with the universal. When particular determinations are grasped
only in their immediacy, they are misconceived as mere singulars or
individuals. In fact, they should not be misconstrued as existing independently
of the universal of which they are momentary aspects. This is the defect of
atomic thinking, or reductionism, which separates the true universality from
the particular, and then makes the particular the universal.

It is this unity of the particular and universal that is the true
individual or the Concept, which when considered in its immediate being as
falling away from the Concept, i.e. as a distinct and externally existing
individual, is the existent individual.